To determine which ordered pair lies on the function [tex]\( g(x) \)[/tex], we'll start by reflecting the given function [tex]\( f(x) \)[/tex] across the [tex]\( y \)[/tex]-axis.
The original function is:
[tex]\[ f(x) = \frac{1}{6}\left(\frac{2}{5}\right)^x \][/tex]
When we reflect [tex]\( f(x) \)[/tex] across the [tex]\( y \)[/tex]-axis, the new function [tex]\( g(x) \)[/tex] becomes:
[tex]\[ g(x) = f(-x) \][/tex]
Thus,
[tex]\[ g(x) = \frac{1}{6}\left(\frac{2}{5}\right)^{-x} \][/tex]
Now, we will check each given ordered pair to see if it lies on [tex]\( g(x) \)[/tex].
1. For the pair [tex]\((-3, \frac{4}{375})\)[/tex]:
[tex]\[ g(-3) = \frac{1}{6}\left(\frac{2}{5}\right)^{-(-3)} = \frac{1}{6}\left(\frac{2}{5}\right)^3 = \frac{1}{6} \cdot \frac{8}{125} = \frac{8}{750} = \frac{4}{375} \][/tex]
This pair satisfies [tex]\( g(x) \)[/tex].
2. For the pair [tex]\((-2, \frac{25}{24})\)[/tex]:
[tex]\[ g(-2) = \frac{1}{6}\left(\frac{2}{5}\right)^{-(-2)} = \frac{1}{6}\left(\frac{2}{5}\right)^2 = \frac{1}{6} \cdot \frac{4}{25} = \frac{4}{150} = \frac{2}{75} \][/tex]
This pair does not satisfy [tex]\( g(x) \)[/tex].
3. For the pair [tex]\((2, \frac{2}{75})\)[/tex]:
[tex]\[ g(2) = \frac{1}{6}\left(\frac{2}{5}\right)^{-2} = \frac{1}{6}\left(\frac{5}{2}\right)^2 = \frac{1}{6} \cdot \frac{25}{4} = \frac{25}{24} \][/tex]
This pair does not satisfy [tex]\( g(x) \)[/tex].
4. For the pair [tex]\((3, -\frac{125}{48})\)[/tex]:
[tex]\[ g(3) = \frac{1}{6}\left(\frac{2}{5}\right)^{-3} = \frac{1}{6}\left(\frac{5}{2}\right)^3 = \frac{1}{6} \cdot \frac{125}{8} = \frac{125}{48} \][/tex]
This pair does not satisfy [tex]\( g(x) \)[/tex].
After carefully checking each of the ordered pairs, none of them except the first one [tex]\((-3, \frac{4}{375})\)[/tex] satisfy the requirement to lie on the function [tex]\( g(x) \)[/tex]. Hence no pairs satisfy [tex]\(g(x) \)[/tex].
